Notes - Uncertainties and Methods - AQA Physics A-Level
AQA Physics A Level Uncertainties and Methods
Notes
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3.1 Measurements and their errors
3.1.1 - Uses of SI units and their prefixes
SI units are the fundamental units, they are made up of: Mass (m): kg (kilograms) Length (l): m (metres) Time (t): s (seconds) Amount of substance (n): mol (moles) Temperature (t): K (kelvin) Electric current (I): A (amperes)
The SI units of quantities can be derived by their equation, e.g. F=ma For example, to find the SI units of force (F) multiply the units of mass and acceleration kg x m s-2 gives kgm s-2 (Also known as N)
The SI units of voltage can be found by a series of steps:
V=
E Q
where E is energy and Q is charge , E=? m v2 so the SI units for energy is kg
m2s-2 (the units for speed (v) are ms-1 so squaring these gives m2s-2 )
Q=It (where I is current) so the units for Q are As (ampere seconds)
So
V=
kgm2s-2 As
V= kgm2s-3A-1
Below are the prefixes which could be added before any of the above SI units:
Name Tera Giga Mega Kilo Centi Milli Micro Nano Pico Femto
Symbol T G M k c m ? n p f
Multiplier 1012 109 106 103 10-2 10-3 10-6 10-9 10-12 10-15
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Converting mega electron volts to joules: 1eV=1.6x 10-19 J
e.g. convert 76 MeV to joules: First, convert from MeV to eV by multiplying by 106 Then convert to joules by multiplying by 1.6x 10-19
Converting kWh (kilowatt hours) to Joules: 1 kW = 1000 J/s 1 hour= 3600s
1kWh = 1000 x 3600 = 3.6x 106 J = 3.6 MJ
76 x 106 eV 1.216 x 10-11 J
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3.1.2 - Limitation of Physical Measurements
Random errors affect precision, meaning they cause differences in measurements which causes a spread about the mean. You cannot get rid of all random errors. An example of random error is electronic noise in the circuit of an electrical instrument To reduce random errors:
Take at least 3 repeats and calculate a mean, this method also allows anomalies to be identified
Use computers/data loggers/cameras to reduce human error and enable smaller intervals
Use appropriate equipment, e.g a micrometer has higher resolution (0.1 mm) than a ruler (1 mm)
Systematic errors affect accuracy and occur due to the apparatus or faults in the experimental method. Systematic errors cause all results to be too high or too low by the same amount each time. An example is a balance that isn't zeroed correctly (zero error) or reading a scale at a different angle (this is a parallax error). To reduce systematic error:
Calibrate apparatus by measuring a known value (e.g. weigh 1 kg on a mass balance), if the reading is inaccurate then the systematic error is easily identified
in radiation experiments correct for background radiation by measuring it beforehand and excluding it from final results
read the meniscus (the central curve on the surface of a liquid) at eye level (to reduce parallax error) and use controls in experiments
Precision Repeatability Reproducibility Resolution Accuracy
Precise measurements are consistent, they fluctuate slightly about a mean value - this doesn't indicate the value is accurate
If the original experimenter can redo the experiment with the same equipment and method then get the same results it is repeatable
If the experiment is redone by a different person or with different techniques and equipment and the same results are found, it is reproducible
The smallest change in the quantity being measured that gives a recognisable change in reading
A measurement close to the true value is accurate
The uncertainty of a measurement is the bounds in which the accurate value can be expected to lie e.g. 20?C ? 2?C , the true value could be within 18-22?C
Absolute Uncertainty: uncertainty given as a fixed quantity e.g. 7 ? 0.6 V
Fractional Uncertainty: uncertainty as a fraction of the measurement e.g. 7 ?
3 35
V
Percentage Uncertainty: uncertainty as a percentage of the measurement e.g. 7 ? 8.6% V
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To reduce percentage and fractional uncertainty measure larger quantities e.g. a longer rope.
Resolution and Uncertainty Readings are when one value is found e.g. reading a thermometer, measurements are when the difference between 2 readings is found, e.g. a ruler (as both the starting point and end point are judged).
The uncertainty in a reading is ? half the smallest division, e.g. for a thermometer the smallest division is 1?C so the uncertainty is ?0.5?C
The uncertainty in a measurement is at least ?1 smallest division, e.g. a ruler, must include both the uncertainty for the start and end value, as each end has ?0.5mm, they are added so the uncertainty in the measurement is ?1mm
Digital readings and given values will either have the uncertainty quoted or assumed to be ? the last significant digit e.g. 3.2 ? 0.1 V, the resolution of an instrument affects its uncertainty
For repeated data the uncertainty is half the range (largest - smallest value), show as
mean
?
range 2
You can reduce uncertainty by fixing one end of a ruler as only the uncertainty in one reading is included. You can also reduce uncertainty by measuring multiple instances, E.g.to find the time for 1 swing of a pendulum, measure the time for 10 e.g. 10 swings in 6.2 ? 0.1 s, the time for 1 swing is 0.62 ? 0.01s (the uncertainty is also divided by 10)
Uncertainties should be given to the same number of significant figures as the data.
Combining uncertainties
Adding / subtracting data- ADD ABSOLUTE UNCERTAINTIES
E.g. A thermometer with an uncertainty of ? 0.5 K shows the temperature of water falling from 298 ? 0.5 K to 273 ? 0.5K, what is the difference in temperature?
298-273= 25K 0.5+0.5=1K (add absolute uncertainties) difference= 25 ? 1 K
Multiplying/dividing data- ADD PERCENTAGE UNCERTAINTIES
E.g. a force of 91 ? 3 N is applied to a mass of 7 ? 0.2 kg, what is the acceleration of the mass?
a=F/m =91/7 =13m s-2
percentage
uncertainty=
uncertainty value
?
100
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Work out % uncertainties
3 91
?
100
+
0.2 7
?
100
= 3.3% + 2.9%
add % uncertainties
= 6.2%
So a= 13 ? 6.2% m s-2 6.2% of 13 is 0.8
a=13 ? 0.8 m s-2
Raising to a power- MULTIPLY PERCENTAGE UNCERTAINTY BY POWER
The radius of a circle is 5 ? 0.3 cm, what is the percentage uncertainty in the area of the
circle?
Area = x 25 =78.5 cm2
area= r2
%
uncertainty
in
radius=
0.3 5
?
100
= 6%
% uncertainty in area= 6 x 2 (2 is the power
from r2 )
78.5 ? 12% cm2
= 12%
Uncertainties and graphs Uncertainties are shown as error bars on graphs, e.g. if the uncertainty is 5mm then have 5 squares of error bar on either side of the point A line of best fit on a graph should go through all error bars (excluding anomalous points)
The uncertainty in a gradient can be found by lines of best and worst fit, this is especially
useful when the gradient represents a value such as the acceleration due to gravity:
Draw a steepest and shallowest line of worst fit, it must go through all the error bars
Calculate the gradient of the best and worst line , the uncertainty is the difference
between the best and worst gradients
percentage uncertainty =
|best gradient-worst gradient| best gradient
?
100% (modulus lines show it's
always positive)
When the best and worst lines have different y intercepts, you can find the uncertainty in the y-intercept, which is |best y intercept-worst y intercept|
percentage uncertainty =
|best y intercept-worst y intercept| best y intercept
?
100%
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3.1.3 - Estimation of physical quantities
Orders of magnitude - Powers of ten which describe the size of an object, and which can also be used to compare the sizes of objects E.g The diameter of nuclei have an order of magnitude of around 10-14 m.
100 m is two orders of magnitude greater than 1m
You may be asked to give a value to the nearest order of magnitude, here you must simply calculate the value the question is asking you for and give it only as a power of ten E.g If the diameter of a hydrogen atom is 1.06 ? 10-10 m, find the approximate area of the entire atom (assuming it is perfectly spherical), to the nearest order of magnitude. Find the area using A= r2 ? (0.53 ? 10-10)2 = 8.82 ? 10-21 m = 1 ? 10-20 m (to 1 s.f) Therefore the area to the nearest order of magnitude is 10-20 m.
Estimation is a skill physicists must use in order to approximate the values of physical quantities, in order to make comparisons, or to check if a value they've calculated is reasonable.
Experiment design Variables - quantities that have potential to change Independent variable - the variable changed in an experiment Dependent variable - variable that is measured in an experiment Control variables - variables that are kept constant in an experiment
Apparatus and techniques Think about what units the measurements will be in and the range over which you'll take them, a pilot experiment (a small scale experiment in preparation) can help plan this so the most appropriate apparatus can be chosen e.g. not using a mass that is too heavy The experiment must test what it sets out to and have a clear method, controlled variables, appropriate apparatus and techniques, and a method that gives precise results and is safe and ethical. Risks and hazards must be identified and safety measures taken e.g. radiation shielding, work ethically by not harming animals and the environment
Data Discrete - only certain values can be taken, e.g. number of objects. Display on scatter graphs and bar charts Continuous - can take any value on a scale e.g. current in a circuit. Display on line or scatter graph Categoric - values that can be sorted into categories e.g. types of material. Display on a pie or bar chart Ordered - data that can be put in ordered categories e.g. low, medium,high. Bar charts can display ordered data
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When data is displayed in a table the first column should be for the independent variable, the next n columns for n repeats of the dependent variable and the columns after that for processing the data e.g. calculations. The data within a column should be to the same number of significant figures. Quantitative data uses numbers whereas qualitative data is observed but not measured with a numerical value e.g. colour.
sum of repeated measurements
mean=
no. repeats
Anomalous results (results which don't fit the trend) should be ignored in calculating
averages.
Graphs Scatter graphs show correlation of data by putting a line of best fit through the points.
If two variables are positively correlated, it means that if one increases in value, so does the other.
Whereas negative correlation means the opposite - if one variable increases the other decreases.
No correlation means there's no relationship between the variables
For lines of best fit on curved graphs draw a smooth curve with about half the points on either side of it (excluding any anomalous results). 2 variables may not be correlated but both affected by a third variable (correlation doesn't indicate causation) e.g. ice cream sales are positively correlated to suncream sales but temperature is the variable affecting them both
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